Gravity Analysis Srsconverges Eh4 D Typed Residual Discrete Torus Family Bridge

The discrete torus bridge

In Recognition Science, a framework where reality keeps a ledger of discrete recognition events, one of the central tasks is to show that the smooth equations of general relativity emerge from this discrete foundation. The declaration typedResidual_discrete_torus_family_bridge is a formal theorem in the framework's machine-checked library that establishes one precise step in this program: it proves that a family of discrete approximations, built on a torus of increasing size, converges to the correct continuum expression for the Einstein-Hilbert action in the weak-field limit.

Specifically, the theorem states that for any non-zero mode on the discrete torus, the ratio of a certain residual quantity to the squared momentum tends to the corresponding continuum value as the torus side length grows without bound. This is a convergence result: it shows that the discrete bookkeeping, when taken to the infinite-size limit, reproduces the smooth continuum formula. The theorem is proved in the Lean 4 proof assistant, and its axiom audit is clean, relying only on the standard three axioms of the ambient type theory.

The bridge is a necessary but not sufficient condition for the full recovery of general relativity. The framework's own documentation is explicit about this: the result covers only weak-field quadratic action convergence. It does not establish the sourced Einstein equation, continuum Ricci curvature or stress-energy, horizons, coframes, arbitrary-curvature general relativity, or the full nonlinear continuation of the action. These remain open targets.

What the theorem does provide is a rigorous foothold. It shows that the discrete-to-continuum bridge does not fail at the level of the quadratic action, which is the first place a mismatch would appear. This is a concrete, checkable step in a much larger program, and it is the kind of result that separates a framework that merely asserts a connection from one that proves it, one theorem at a time.

/-- **THEOREM (R4):** discrete torus bridge inhabited (uses R2). -/
theorem typedResidual_discrete_torus_family_bridge :
    TypedResidual_discrete_torus_family_bridge :=
  discrete_torus_family_bridge

theorem typedResidual_discrete_torus_family_bridge_closed :
    TypedResidual_discrete_torus_family_bridge :=
  typedResidual_discrete_torus_family_bridge

What this page does not claim

The theorem does not prove the full Einstein field equations, only the weak-field quadratic action convergence. It does not claim to recover the sourced Einstein equation, continuum Ricci curvature, horizon physics, or arbitrary-curvature general relativity. The theorem does not establish the physical recognition-to-linking bridge that connects the discrete ledger to spacetime.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Gravity/Analysis/SRSConvergesEH4D.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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